Filtering circuits



Jan. 13, 193,1. E. L. NORTON 1,788,538

FILTERING CIRCUITS Filed April 16, 1929 2 Sheets-Sheet 1 A F16. 4 B L Il ll l z i MM Q,

FREQUENCY //V VE N TOR E L. Non ro/v By y MURA/EV Jan 13, 1931. E. L. NORTON 1,788,538

FILTERING CIRCUITS Filed April 16. 1929 2 SheetS-Sheet 2 I I I I I I I I I I L6 l I I I l I Patented Jan. 13, 1931 @rnc-E j amuse-DL. NonmoN, oFV Elisir cimiteri,` NEWl quasar, essienon fro BELL rnlinrnoun LABoRAToRIEs INCORPQRATED, F NEWv YQRK, N. Y., n yc,oleron/,si'rioN,QF NEW f i appncatvionwiea Api-n is,

This invention relates to filtering circuits er and more particularly Vto .Circuits Vhaving u through a. selective circuit. Y

heretofore known have the basic form'ot` a*l network, orartificial hne, of recurrent structure, the branches ,of which are composed oi discrete reactance elements of different kinds,

Such circuits have the property that, ifininitely extended, or if terminated "bylan iinpedance device equivalent to an infinite eX?4 tension, oscillations within certain ,ranges of irequencies can he' transmitted "throughl Loi other frequencies are strongly attenuated-` Since the impedance of fthe"'infinitely ex-.

tended yline is variahlewith yfrequency it cannot ordinarily hesimulated bythe terminal y .device in which the selectedoscillationsare to4 he i1-sed, hut within the transmission hands the impedance is resistive and, lfor 'aliinited range, is fairly constant, vsc tliiat the simula.l tion may be approximated-by a resistance termination. DueV to the-,disparity of the resistive load and the characteristic impedance oi"- t-he line, reflection effects are lpro duced which give rise toirregularities in .the transmission characteristic of the circuit in the formof undulati0ns, the amplitudes of which are greater the greater the impedance disparity. These' undulations may alsoibe attributed to resonances of the selectiveciri cuit, the resistive-termination being inade-V V quate for complete darn ing. For many purposes the resonance e ects are of little importance Since their magnitude is greatly diminished bythe energydissipation in i,the

impedance elements, but in other cases, "wherev`v the internaldissipation 1s ver-y small or where i the operating conditions require a high dej gres of uniformity, theirregularities dueto undainped resonance may' heobJectionable.

ln accordance with this invention'loroadV band selectivesystems are provided'which are i'ree 'from transmission irregularities of the are deserted faber-e f The-.Systeme @Qm- Broad hand selective circuits lot the types.

FILTER'ING Y CII-licores 19:29; serial No. aaeoaj I prise artiiiciallines l.ofthe series-shunmor 'ladden type whichv 'have generalV resemhlances, in form,to well `known broad band iilter circuits, butuwhiclidifer therefrom in f the values assignedto `the impedance elements. The successive sections of the arti-vr iciail linesa're not uniform and the impedance elements are so chosenthat the absolute rnagnitrudes of the received oscillations, Jfor l a constant inputivary with frequencvin accordance with a'uprescribed law. Thelaw of variation is, such that Athe received oscillations i eoL have ainaximumvalue atjonefrequency in'l Aany pass'balid, and fall off ,Continuously ffOm ,this maximum as ,thelfrequny 'changea the rate of' falling O? being Very slow atiirst Y e and great at the ,limits -Of the bendS'X-d i them withoutr-attenuation whileoscillations pressed mathematically, the law of the .variavtionis. p Y

y ,s (l) y where. [J denotes the magnitude of thereceived oscillations, iM is a constant, n `i'sthe f lui number ,of impedancehranches in the artiicialline, and X is an imaginary `odd rational unction ovthe frequency. Inthis way a smoothyi-,transmissionfcharacteristic is obtained, :free from undulations, and very uniform" over the whole transmission range.

The Circuits ofthe lwenterll may be sels@- tive to a. Isingleleroad band ofirequencies,

as are low-pass, high-pass,` and hand-pass iilters, or they may be selectiveto a plurality 4,Gf bands, @wending t0 v*die Complexity 0i the structure lofthe branch impedances. lneach ot thehands thetransmission characteristic has the e'aturles mentioned above. l

'series branch impedance and'any shunt branch impedance fis a constant, invariable with frefuieilffyf I e The Principles .of the invention ,erle .applicable .alike ,the electric and in kthe mee l chanical systems the specified law of varia-A chanical field. In the electrical field the received oscillations to whichEquation 1 applies may be current or voltage oscillations, or may represent the charge in a condenser, and the constant input may be eitherV constant currentor constant voltage. Similarly in metion may applyto vibratory velocity or vibratory force in response to a constantY input either of force or of velocity. The invention is here described in relation to electrical circuits, a limited number of its embodimentsV OtherV species of the invention as applied Vto me-4 chanical systems are disclosed in my copend- .Y ing application Serial No. 367,487 filed-May being illustrated by the drawings.

` Referring to the drawings, Fig. 1 shows aschematic arrangement of one general form of the invention;

Figs. 2, ,3 and 4; illustrate particular cir cuits of the general'type shown in"Fig. l,`

adapted for single'band selection; f Y

Fig. 5 illustrates the structure of a multilbandV selective circuit according to the in-Y vention of the gass bands;

Fig.

tion; y Y l.

Figs. 8 and 9 illustrate additionalforms ofthe invention; and

Fig. 10 shows the invention appliedrto an' amplifier circuit. u

The type of structure shown in Eig. 1 has the property that the received voltage in re-v sponse'to a constant limpressed voltage varies with frequency in the manner prescribed by n Equation 1. The circuit comprises a terminal resistance R, representing an energy ab-y sorbmg load, and a series-shunt artificial-line, h

having series bran'ch'im'pedances alXR, aSXR, LXR, and shunt branch impedancesv t .a R agX MX en lX The subscripts of the coeiicients al, a2., etc..y denote the order ofthe branch counted from. the terminal resistance, there being in all n,

branches and the fnth branch being in series. The input voltage is'impressed on the terminalsrAA by a source 11, andthe received voltage is that between the terminals BB of the resistive load. The factor X is an imaginary odd rational function of the frequency, the general form of which will beVY discussed later. Its particular form in any given case depends upon the nature and arrangement of the impedance elements 1n ,the branch. The coeiiicients al, a2, etc. are

numerical factors which varyprogressively4 from branch to branch, their values being Fig; 6 is a diagram illustrating the locationVK shows the type of transmission char- Y v acteristic peculiar to the systems of the inven-v determined in accordance with the prescribed form of the transmission characteristic.. The branch impedances are wholly reactive, this being indicated bythe imaginary character of the factor X. They maycomprise single Vinductances or capacities,or they may consist of resonant combinations of any degree of complexity.V The series and' the shunt impedances, as'lindicated bythe positions of the factor X, are inversely related with respect to vtheir frequency variation, so that the product of any pair is invariable with frequency. The design of inversely related impedancesis described in U. S. patent to O. J. Zobel, No. 1,509,184, issued September 23, V1921i-, which also describes uniform' wave "filters, 'having an inverse relationship beytween the series and the shunt impedances.`

- The determination of thenumerical values f of the coeflicients al, abete. is a lengthy and involved mathematicalprocedure which need not be given in Vfull here,l but which ywillV be 1 y y VAMX?+A 1X+ 1 y(2) iniwhich the coeticientsAo, A1, etc. are vfunctions of the coefficients 11,-a2, etc. of the imi 'pedance branches. Thesimple character of the power series is due to the fact that the .Y series impedances and the shunt admittances are all proportionalto the same frequency function X, this permittingthe ratio to be expandedas a function of a single variable.

Equation 2 may be written kin the form K where X1, X2, etc. are the fn, roots of the eX- pression on the right of` Equation 2, and the desired ratio, namely that of thef. absolute magnitudes of E1 and E2,can'be found by multiplying the expression on the right of -If now, it is stipulated that the absolute magnitude of the voltage ratio shall have the value.. Y Y

E,2 2. Y i E-2 X +r 5) the identity is` established that incassa `X12, etc. respectively equal to the-n, ath

roots of il, the sign being taken if n is odd and the sign if n is even. The values of the roots X1, X2, etc. may be written as when n is even, and

i when n is odd, the sign being chosen so that the real part corresponding tothe damping constant, is negative.l 1 i These roots define the character of the transient, or free, oscillations of the circuit. The

imaginary parts of the roots areproportional to the natural frequencies and the real parts y j to the corresponding damping constants.` If

plotted in the complex plane with the real parts as abscissae and the imaginary parts as The roots in their complex Aform are someordinates, it will beseen that the roots are represented by equal vectors spaced at equal angles, the angle between successive Vectors being equal to times termed the complex 'damping constants of the circuit. A. circuit for which the complex damping constants have the values indi- ,i

cated above possesses the unique transmission characteristic expressed ,byl Equation l. It is notfree from natural or` transient oscillations, but, in its response to forced oscillations, resonance effects at the several natural periods are not in evidence. The response to forced oscillations resembles that of asingly resonant system but` is much broader and Hatter. -In the more complex systems of the invention a plurality. of transmission bands may exist, the response characteristicsin each i band being of the character of a single broad resonance, free from irregularities at the natural periods. The roots in such cases each def vfine a plurality of natural periods, one Vfor each 'transmission range.

The more x1, x2, having been ,forma-u1@ l cceiicients A1, A2, etc. in Equation 2 can be determined by standard mathematical processes. These coefhcients are related to the roots in the following manner:

isthe sum ofthe roots;

il lA,

.is'the sum of the i tim`e,'and soon.l

ratio, the next step is to determine the values' fof the impedance;cofhcients al, c2, etc., so

` be found.

Ycase where the V'number of given'. y n The frequency factor X is evidently'a simi-V .lar function `vto YV'that which defineswthc fre- Y Ais the sum ofthe products of the roots taken twoatatim'e;V "i 'rf 1W "Anf:` v

products taken `three Y at a c Having determined the necessary values of the coeiicien-ts'o-f Equation 2 tomake it correspond to the stipulated Vform of the voltage that v'the physical*networkxwill have the prescribed-characteristic. To d0 rthis Equation .2 maybe rewritten with the coeiiicients A1,

A2, etc., fuily expanded, that is, expressed in' terms of the impedance coeci'e'nt's al, a2, etc., of which they are functions. Two expressions are thus established'for the ratio soA matter, but it invclvesonly standard algec,

hraic processes andy icads to asimple result-,f`

vThe values ofthe impedance cceicients are found to be as follows: f

Sai-f i 1f 2n sin .2n

.2n @sin 2712;?" ,(8)

'Cr-l' l aff-1 V l `aan-nem 2n .Y

These values arereadilycomputed infany the branches quency variation of the series branch imp edances,` orV of. the shunt branch vadmite tances. A* r:[t'is also, -necessarily,arfrequency ratio having dimensions of a simple numeric, since the other factors of `the expressions for the branch impedances are the numerical coeifcints al, a2, etc. and the ne.

` sistance R. VLVVherethe branch impedances are of simple types the form yof the-factor X can be ascertaind by inspection; thus, for example, if the series branches consist of simple inductances,rX will be directly proportional to frequency.; vWhere the :branch impedances are complex reactive structures, involving several resonantcombinations, the series branch impedances can be expressed by the general formula' Y in which S is the reactance, H isfa constant, f1, f3, etc. are the successive frequencies at which the impedance is zero, and f2, i, etc. the successive'frequenciesat which the impedance is iniinite,.tliere being in all 2p-1 uated, the degree'of attenuation increasing very rapidly as thenumber of the impedance branches is increased.` For tions is .707 of the input amplitude, and .for lower values of lXl the ratio is substantially unity. The frequencies for which [X] is unity are therefore in the nature of cutoff frequencies,

foi-m transmission. The transmission bands are centered roughly about the frequencies for which \Xl is zeroand the attenuation ranges about the frequencies for which is infinite. v v

The design of circuits to transmit oscilla-e tions between prescribed limits of frequency may be carried out in a direct manner by giving the series branches an appropriate structure and assigning such values to the elements that the reactances have the magnitude taR at the specified limits, the coeicients a being that appropriate to the branch chosen.

generally simpler.

The design of the shunt branches follows from the reciprocal relationship involved. However the following procedure, which is based on the design of uniform wave filters is A uniform line of recurrent structure, having series branches of impedance 2XR and shunt branches of impedance lXI equal to.

unity the amplitude of the received oscilla-V marking the limits of uni-V l has its'transmissionband 4limits defined by the condition y Y :Ej

which is the same as the condition'deterlV mining the band limits inthe circuits of the inventionf- Suchia line, likey the circuits of the invention, is characterized by an inverse relationship between its' series and itsL shunt i1npedances,the product of-any pair of these" impedances being constant and equal to is described l inV U; S."Patent11,227,113, issued May 22 1917, to G. A.l Campbell, and in U.,S f

The design of wave filterscf this type, to which Vthe nameV constantK has been given Patent 1,509,184. issued September 23, *1924, y

to O. J. Zobel. The design formulae of these patents,mayl be usedto'icoinpute the imped! ance elements of a typical section of the constant K line having the "prescribed cut-off frequencies, and, from the inipe'dances of this typical section, the designV` of the' corresponding circuit of the invention may be computed by virtue of Y fthe v relationship mentioned above. The'impedances of the series branches are Vequal to half the seriesl inipedanceiof the related constant K sectiony multiplied by the appropriateV impedance coefficients -of Equation 8, and the 'impedances of the shunt branches are found by taking twice the shunt 2 impedance of thev iterative structure and applying the appropriate coefiicients.

`The application ofthe foregoing principlesA will be illustratedin connection with Figs. 2, 3, and 4,' which Vrepresent specific` circuitsfof theeinvention' adapt-ed topass a single band of frequencies. In these circuits the lawpofEquation 1 applies to the output voltage for a constant voltage input, the characterizing feature beingy that the circuit ter#` minates in a `series branch at the input end. The ratio of the input to theoutput voltage is that expressed by Equationl 5 `and, if the sourcewof E. M. F. is of very low impedance, the output voltage will have the prescribed variation for a constant generated E. M. F.

Fig.l 2 represents a circuit having' a lowpass characteristic, the series impedance consisting of simple inductances L21, L25v and L25, and the' shunt impedances comprising simple capacities C22 and C21.. The number of branches is five, and the corresponding impedance coefficients are al: 309 a2: 896 a3: l. 380 ai: 1. 695 f. as: 1. 545

`Denoting the cut-off frequency by fo, the Y having Vthe .seme @bend .lmit are elven Tby The desired follow:

` Y .U2- a2 'OME-Tm' 0.24.- zqrfzef Y The frequency factor X has the value and the voltage `transniiission characteristic is defined by d v v E2. "/if ,0) (11) y Y The .form of the transmission characteristic of .the 10W-pass y.typeof circuit is` shown Vin Fig. 7 in Whchthe magnitude of the Voltage ratio is plotted )against .thefrequency function .yf/7%.v The `,uniformy Acurves 12, 113, and

1 4 representl the characteristics V.of circuits v having 3,:5 and 7 branches IGSPQCtl/'ly 11h-lsf trating the highdegreeof uniformity. that exig.- 3 represents a circuit ;of the pass type,\ adapted to transmit oscillationsof p all frequencies above the `.assigned limitffo. Y

The frequency `-fact'orfin lthis y(':aselis The prototype-iterative structurewcornprises series capaceities'Cl'L andshuntyi,Iiductan'ces Y i i having` the Values" iinpedance i values immediately"` f and' semi 'y The branchradjacentthe resistance and are determined, Eas follows frm which the @nella branch "impedaneslfo p R', is in shunt ratherthan inlseriesia's in vrigs. 1, and 2, but

sign of thecirouit.v e l ",*The circuit ofiFig. 4 is adapted to pass athishasno effecton the operation'or the de- ,l

' band .of frequenoiesbetweenpresented finite 1imits,fdented^by f1 and f2. The series branches comprise simple resonantfoircuits, p all resonant at the same frequency, and the* shunt branches4 compriseparallel circuits' ant1-resonant at thesame frequency. The frequency factorhasthe Value I m 2 v The-values of the impedance elements in the prototype iterative structure, are

vWhereV Y 'and 'i i L1; `for the shunt' loranohesjr'lhesef'lead'to lthe Values `of the impedanoeelementsof thejcir andjSfOTOIll. 'i f adepte@ .mgltebandgtiensmission, eg.-

first branchythat is,v theVE ample of a structure of this type being shown" in Fig. 5. Only a single section is illustrated, but this suffices to show the character of the branch impedances; The particular struci ture shown has three pass bands, one of which Y starts at zero frequency The series impedance Z.rl comprises an inductanceand two anti-resonant circuits, allin series. This coin- `bination is resonant at three frequencies, in-

'cluding zero, and frequencies, including infinity. The shunt is-anti-resonant at three impedance Zb'is inversely related to Za, being anti-resonant at the resonance frequencies of Za and viceversa. It comprises a condenser of X between ;-1 and :t1 and are indicatedV and two series resonantcircuits all connected in parallel. are Ylocated about the three resonancefrequenciesl of the series branch impedance, and may be placed roughly in their desired positions in the frequency scale by properlyplacing the resonance frequencies Additional bandsmay be obtained byKadding-anti-resovnant circuits to the series branch and correspending resonant circuits to the shunt branch. The illustrated by Fig. 6, which shows the form of the frequency factor X fortlie circuit of Fig. 5. The value of increasesfrom zero at the first resonant frequency, 72:0, to infinity at the rstl anti-resonant frequency` f2 infinity at the' and passes through zero and successive frequencies f3, f4, f5, and f6=oo.

The transmission ranges are defined by values in the figure by the thickened portions of the horizontal axis., n

frequencies placed according to the. desired band locations. A plot of the impedance will indicate the positions ofthe band limits and, if these are V incorrect, the constants .may be variediin successive"trialsuntil a good approximation to the desired'positions is obtained. The; designof thel shunt impedance Y i f follows from the'inv'erse relationship, and the final design is obtained by applying the impedance coefcients` Aal', a2` as already indicated. c

The types of circuit thus fari; discussed have Vthe propertythat the receivedvoltage across the terminal V.resistance .bears the ratio toY the .voltage impressed'ontheinput terminals defined byEquation. "'.Since the` terminalV impedance is fa resistance it'followfs that theV The three transmission 'bands formationv of the pass bands is Y :ter circuit impedance, y so y fcurrent may have the prescribed variation to proceed'towards the final vvalues given rby' Equation current therein 'hasthe same variation as the received voltage. thesev circuits Itfollows then, that for T. #Rv *1+ e le where I2 is thev current in the resistance R."

Mor'eover, ,it 'follows froin.- the-principle lof reciprocity that, if the wave source .be inserted in series with the resistance'R, the terininalsy AA being short circ'uited, the current in the final imp'edance'branch, anXR of Fig.

1, will likewise have the value given by Y Equation 15.

A'modiied form of the invention is illustrated in Fig. 8, the circuit arrangement ofthis form being suchthat the ratio of the input current to the output current follows the prescribed .,law. that of Fig. 1 in that the lnth impedance The circuitv differs from branch is in shunt insteadof in series. As

in the other general form of the invention,

vthe first branch, adjacent the terminal resistance R may beeither in series or in shunt.

Denoting the input current and the output current by I1 and l2 respectively, the transmission characteristic is expressed by I 2Y Y X, and n, having the same significanceas before.- p

.[In'thefigure a resistance R0 isshown connected in series with the' source. highfresistance inserted` to `make [the input This' 'is a current substantially independent of the filthatV the c output vfor a constant generated E. M. F. c `In certain A cases where thefinputffcufrrent is rendered independentoffthe filter impedance by other circumstances the;resistanceR5y not neces'- sary.` Thefcoefficients ai,?a2, etc.'have the The circuit of Fig. y9 is'of the same'type as thatfof v.Fig-'8.,l characterized bythe fnth branch'being connected'in-shunt. Thev wavev sourcei11 is yconnected in seri-eswithvthe terminal resistance R `and the terminals {AA are openfcircuited'. With this arrangement the ratio of the E. M. F., El of the source 1l, to the E. yacrossthe nth branch, that is between te'rininalsAAf is given by This nsiiowisv from intimiteit by the prin! ciple of reciprocity.

lnaFiglQzis shownthe, application ofthe circuits: QfFigS. 8 and 9;;to`fan amplifier ,couv pling circuit. l TheV circuit'yis'of the .single band-pass type and is arranged to operate between the finite later resistance of one vacuumV tube ampli fer 16, and the subst'an-Y iio iis

ing amplifier 17. The shunt branches of thecoupling circuit, Z1, Z3, Z5v consist of parallel combinations of iiiductance and capac-- ity, and the series vbranchesZ2 and Z4 are simple series resonant combinations. The values of the impedance elements are computed as for the circuit of Fig. Ll", the resistance R being the vplate circuit resistance of amplifier 16, and the coemcieiits al, a2, etc. being applied respectively to the branches Z1, Z2, etc., in the order indicated bythe drawing. The coupling circuit, as it is shown, does not' introduce any voltage amplification, but this may be provided by the insertion of a suitable transformer or by incorporating a transforming effect in the circuit itself iii the manner described'in U. S. patent to Norton, No. 1,681,554, issued August 21, 1928. 'Y

lVhat is claimed is: n 1. A frequency selective circuit comprising, in combination with a resistive terminal device, a Wave transmission line having al` frequency selective ,transmission character-- istie, and the values of the impedances `being proportioned with respect to the resistance of said terminal device and to the number of impedance branches in the line, whereby y the oscillatory responseat one end ofy the line for a constant oscillatory input at the other end is substantially proportional tol n www where n is the number of impedance branches in the line and X is an imaginar odd b' y a series shuntartifcial line and a resistive terminal device connected at oneend thereof,

rational function of the frequency. 2. A frequency'selectivecircuit comprising la. four-terminal network having n branches, and a resistiveV terminal device connected to one pair of terminals thereof, the branches of said network confiprisingl reactive impedances of dissimilar types, and the iinpedaiices of the branches being pro-l portioned with respect to the resistanceof said terminal device,- to the number of branches, and to prescribed frequencies de.-V

fning transmission band limits, whereby the magnitude of the oscillatory response at one pairof terminals of the networlrfora conu` stant `oscillatory input Vat the other pair isfsubstantially proportional t`o w/i+IXi2n where X is an imaginary odd rational func- 3. frequency selective circuit comprising;

dances f aidf series-Br.

alternately inseries and i shunt" lies being proporfrequency function, ,and" s. varying progressively' from aff afresis'tiye terminal pedance Vcon *cted one? end Ythereof,,saidl une comprising reactive'impedancebraadiesf seaiprriefsjectiioa whereby'ihearpiiufdee of* the frequency variation ,off ,the series dances and the,rshuntaldnittances.'

tlieadmittances" f een', and@ aufaefipaifacars alla, @n n seam@ @nid whe@ the .impedance branch' coiintingzfrom;v tvliefiic'i-V 2 am cos 5. A frequency selectivecircuit comprising the series branches of'said line comprising similar reactance" combinations having impedances proportional lto a common frequency function andthe shunt branchesV hav- 'to section in opposite manners respectively, whereby the transmission characteristic isv Vrendered free from irregularities due to resonances of the system;

minal device connected to one pair of termirality of reactive impedance branches of dissimilar types whereby the circuit has a plurality ofnatural frequencies ofVv vibration7 and the impedances of said branches being iio nais thereof, said network comprising a p luw proportioned with trespect to' the number off branches and the resistance of said ,termi- 12oY I 6. A frequency selective circuit comprising -f la four-terminal network and a resistive'ternal device so. that the natural frequencies of v the circuit are proportional tothe sines of multiple ofthe angle 11j/2n, Vwhere n the number of branches;

7. Afrequencyselective y Y a fourterminal network and a resistive terminal device connected toone pair of termi nais thereof, said network comprising a plu#V rality yof reactive impedance branches of dissimilar types whereby the circuit hasl a plurality of natural'frequencies of vibration, the

natural frequencies being spaced in propor- Y tion to the sines of the multiple angles of vr/Qn, where fn, is the number of branchesin the network, whereby the response of the circuit at one pair of terminals to a constant oscillation inputl at the other pair has the character of a single broad resonance in each of its transmissionV ranges. l v

8. A frequency selective circuit comprising a four-terminalv network and a resistive terminal device connected to Aone pair of terminais, said network comprising aplurality of with respect to the number Vof branches and V Y I reactive impedance.v branches of dissimilark types whereby the circuit 'has aplurality of natural' frequencies .of vibration, the impedances of said branches being" proportioned to the resistance ofV said terminal device lso thatvthe CQmPlX. damping constantgpf the r. v circuit are represented b a series of equal vectors in the complex p ane spaced at uniform angles equal to rrr/n where n is the number of branches.

In witness whereof, I hereunto subscribe my name this 9th dayof April, 1929. f

, -EDWARDL NORTON.

incasso circuit comprising 

